Rational Exponents for Generalized Tur\'an Numbers

TL;DR

This paper proves that for any rational exponent between v-1 and v, there exists a finite family of forbidden graphs such that the maximum number of copies of a fixed graph H in an n-vertex graph avoiding this family grows as n to the power of that rational number.

Contribution

It extends previous results by showing that for any rational exponent in a specific interval, a corresponding finite forbidden family can be constructed.

Findings

For every non-empty graph H with v vertices and rational r in [v-1,v], a finite family F_r exists with ex(n,H,F_r)=Θ(n^r).

The result confirms the rational exponents conjecture for a broad class of exponents.

The work generalizes earlier findings by Bukh and Conlon to all rational exponents in the interval.

Abstract

The generalized Tur\'an number $\text{ex}(n,H,\mathcal{F})$ denotes the maximum number of copies of $H$ in an $n$-vertex graph which contains no copies of any graph in a family $\mathcal{F}$ of graphs. The generalized rational exponents conjecture states that for every rational $r\geq 1$ there exist graphs $H,F$ such that $\text{ex}(n,H,\{F\})=\Theta(n^r)$. We extend a result of Bukh and Conlon to show that for every non-empty graph $H$ on $v\geq 2$ vertices and every rational $r$ in the interval $[v-1,v]$ there exists a finite family $\mathcal{F}_r$ such that $\text{ex}(n,H,\mathcal{F}_r)=\Theta(n^r)$.